# Algebra, Geometry, and Cryptology Seminar

## Spring 2019

Organized by Zach Teitler <zteitler@boisestate.edu>.

Time: Fridays, 3:00-3:50
Location: MB124

View archives of past semesters

• Taking the seminar for credit
The seminar may be taken for credit as Math 498 or 598. For information about receiving credit, contact zteitler@boisestate.edu.
• Everybody is welcome!
Everybody interested is welcome to attend and participate! Enrollment for credit is not required. We welcome students to attend and present in the seminar. Attending seminars and colloquium presentations is an excellent way to learn about research topics and senior thesis topics.
• AGC Seminar mailing list
Announcements of upcoming seminars are sent by email to all interested participants. To be added to the AGC Seminar mailing list, contact zteitler@boisestate.edu.

Planning meeting

### January 25

Topology seminar
Jens Harlander
The Corson/Trace characterization of diagrammatic reducibility

### February 1

Topology seminar
Jens Harlander
The Corson/Trace characterization of diagrammatic reducibility, 2

### February 8

Zach Teitler
Hurwitz’s 1,2,4,8 theorem via linear algebra

A product of sums of squares can be written as a sum of squares, in some cases. Identities for the cases of sums of 2 or 4 squares play key roles in the study of which integers are sums of 2 or 4 squares, and are closely related to the complex numbers and quaternions. Cayley discovered a similar identity for sums of 8 squares, closely related to the octonions. Despite wide efforts, no similar identity was found for sums of 16 squares. Eventually Hurwitz proved that in fact, certain identities involving products of sums of n squares are possible only for n=1,2,4,8. This implies that generalizations of the cross product (on R^3) are possible only in dimensions 1,3,7; and real normed division algebras exist only in dimenions 1,2,4,8. We’ll sketch a proof of Hurwitz’s theorem using “elementary” linear algebra.

### February 15

Scott Andrews, Boise State University
Hurwitz’s 1, 2, 4, 8 theorem via representation theory

If n = 1, 2, 4, or 8, then there is an identity expressing the product of two sums of n squares as a sum of n squares. In 1898 Hurwitz proved that these are the only values of n for which such identities exist. I will prove Hurwitz’s result using representation theory. This is a continuation of last week’s talk, however no knowledge of that talk or of representation theory will be assumed.

### February 22

Zach Teitler, Boise State University
What’s a sheaf?

I’ll describe fairly informally what a sheaf is and why they are important tools in geometry. Sheaves attach local data, such as vector spaces or abelian groups, to a topological space. So, in one direction, sheaves give us information about the underlying topological space. In another direction, sheaves give a framework to talk about continuously varying data. We’ll meet stalks. If time permits I’ll at least mention sheaf cohomology. This talk will strongly emphasize the role of sheaves in geometry, as opposed to logic, where sheaves also play an important role, but I will not be able to say anything.

### March 1

Mitchell Scofield, Boise State University
Thesis defense: On the Fundamental Group of Plane Curve Complements

Given a complex polynomial $f(x,y)$ we study the complement of the curve $C$ defined by $f(x,y)=0$ in the complex $xy$-plane. The Zariski-Van Kampen theorem gives a presentation of the fundamental group of this complement. One ingredient of this presentation comes from understanding the monodromy of a fibration induced by the projection of the complex $xy$-plane onto the $x$-axis. This is hard to come by in general. It turns out that under special circumstances the presentation can be computed directly from combinatorial and visual (real) information on the curve $C$. The method in these special situations is similar to the computation of the presentation of the fundamental group of a knot complement in 3-space.

### March 8

Topology seminar
Stephan Rosebrock, PH Karlsruhe, Germany
On the asphericity of labeled oriented trees

### March 15

Morgan Sinclaire, Boise State University
Thesis defense: Formally Verifying Peano Arithmetic

From the beginnings of mathematical logic, a central concern has been the consistency of the axioms that mathematicians assume in their work. In particular, logicians came to study the consistency of arithmetic, and Hilbert’s program of the 1920’s has come to be associated with the problem of proving the consistency of Peano arithmetic (PA). In 1930, Godel’s 2nd incompleteness theorem quickly became regarded as having shown this goal impossible, yet in 1936, Gentzen’s consistency proof gave a surprising but controversial counterpoint. This work is concerned with implementing a version of Gentzen’s proof in Coq, an interactive theorem prover. Although our project is still unfinished, this will show that Gentzen’s proof, as stated, is computer verified. However, the assumptions Gentzen relied on are ultimately up to humans to judge.

### March 22

No meeting this week (Spring break).

### March 29

Sean Ippolito, Boise State University
A Survey of Persistent Homology

Algebraic Topology via homology gives us a framework for dealing with continuous objects in a discrete manner. The pressing question is ‘given discrete data, can we recover the topology of such data?’ The answer is yes, and in this talk I will show how. We will walk through the theoretical foundations of persistent homology, talk about its related tools and extensions such as barcodes, and briskly tour through some applications in seemingly disparate fields of mathematics.

### April 5

Topology Seminar
Uwe Kaiser, Boise State University
Obstructions to ribbon concordance

Topology videos

### April 19

Steven Bleiler, Portland State University

### April 26

Liljana Babinkostova, Boise State University
Strongly Non-zero Points and Elliptic Pseudoprimes

We introduce the notion of a strongly non-zero point and describe how it can be used in the study of several types of elliptic pseudoprimes. We present some probabilistic results about the existence of strong elliptic pseudoprimes for a randomly chosen point on a randomly chosen elliptic curve. This is a joint work with D. Fillmore, P. Lamkin, A. Lin, and C. Yost-Wolff.

### May 3

No meeting this week.